Expeditions were generally provided for by the government, and the expectation was that the explorer would report his findings to the sponsor as a matter of course. Generally speaking, the explorers tended to be rewarded after return in relation to how lucrative their discoveries seemed to be, so they had every incentive to spill out the details (and garnish them a bit, too).

There are some recorded instances where the explorers left behind documents on cairns or other markers visible to the sea in case they would be lost. The Portuguese king João II in fact ordered his sea captains to take along limestone route markers known as padrões that were to be mounted on prominent headlands where they could be easily seen by passing ships.

Usually the expeditions were so small that there wasn't much point of tagging along a separate dispatch ship. Therefore, the news generally had to wait until the expedition returned. In some cases, particularly in the expeditions to India, the explorers were able to send word of their process through e.g. returning merchants while still continuing their voyage, but in most voyages this obviously wasn't an option.

What happened after the explorer returned varied greatly. The Portuguese were very quick to capitalize upon the discovery of a sea route to India. Similarly, the Spanish began to settle the Americas fairly soon after Columbus returned. On the other hand, Sir Walter Raleigh explored the coast of what is present day Virginia in 1584, but the first colonists were sent only 22 years later.

Of explorers dying after grand discoveries, I can think only of some polar explorers like Scott's party, but that was in the turn of 19th and 20th centuries.

Primary source for the above is "The Explorers" by Richard Humble et al., 1979. Also Parry's "The Age of Reconnaissance," 1963.